Supply-side equilibrium : growth and nonpure competition

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Supply-Side Equilibrium:
Growth and Nonpure Competition

The Library of the
MAY 1 b iwi

University of Illinois
of Urbana-Champalgn

Hans Brents

Department of Economics

Bureau of Economic and Business Research

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

BEBR

FACULTY WORKING PAPER NO. 91-0124

College of Commerce and Business Administration

University of Illinois at Grbana-Champaign

March 1991

Supply-Side Equilibrium:
Growth and Nonpure Competition

Hans Brems

Department of Economics

University of Illinois

1206 South Sixth Street

Champaign, IL 61820

HANS BREMS

Box 99 Commerce West

1206 S. Sixth Street

Champaign, IL 61820

FAX (217) 384-0014

(217) 344-0171

1103 South Douqlas Avenue Urbana. Illinois 61801
March 11, 1991

SUPPLY-SIDE EQUILIBRIUM: GROWTH AND NONPURE COMPETITION

By Hans Brems

Abstract

The original monetarist and New Classical supply-side equilibria
remained static models of pure competition. Their capital stock
remained frozen rather than being optimized. At such frozen capital
stock and a "natural" rate of unemployment, physical output, called the
"natural" supply of goods, was Immune to monetary and fiscal policy.

Within a framework of growth and nonpure competition we restate
the supply-side equilibrium and find physical output no longer Immune.
Lowering the rate of growth of the money supply or lowering the tax rate
will raise the equilibrium levels of capital stock and output.

I. INTRODUCTION

Monetarist [Friedman (1968)] and New Classical [Lucas (1972),
Sargent (1973), and Sargent-Wallace (1975)] supply-side equilibria were
purely competitive and as static as the Keynesian demand-side
equilibrium had been: nothing moved. Capital stock remained frozen
rather than being optimized. At such frozen capital stock the physical
output corresponding to a "natural" rate of employment was called the
"natural" supply of goods and was immune to monetary and fiscal policy.

But throughout the last half of our century, inherent dynamics
kept creeping into macroeconomic theory, at first only as afterthoughts.
The Phillips curve took the derivative of the money wage rate with
respect to time. The government budget constraint took the derivatives
of the money and bond supplies with respect to time. Growth theory went
beyond such afterthoughts and took the derivatives of all its variables
with respect to time.

For our growth-theory background we use Solow's 1956 model — old,
simple, and robust and still good for a Nobel Prize. To it we add
government and nonpure competition to see how much of a supply-side
equilibrium will survive.

II. THE MODEL

1. Variables

C = physical consumption

D = demand for money

G = physical government purchase of goods

g = proportionate rate of growth

I = physical investment

k = present gross worth of another physical unit of capital stock

k = physical marginal productivity of capital stock

L = labor employed

n = present net worth of another physical unit of capital stock

P = price of good

R = tax revenue

r = before-tax nominal rate of interest

p = aftertax real rate of interest

S = physical capital stock

w = money wage rate

X = physical output

Y = money national income

y = money disposable income

2 . Parameters

a = joint factor productivity
a,R s exponents of a production function
b = a demand parameter
c s propensity to consume

ti = reciprocal of Marshallian price elasticity of demand
F = available labor force
X 2 "natural" employment rate
M a supply of money

m 2 reciprocal of the velocity of money
T 2 tax rate

All parameters are stationary except a, F, and M, whose growth
rates are stationary.

3. Definitions

Define the proportionate rate of growth of variable v as

9v

dlogev (1)

dc

Define investment as

I'9sS (2)

4. Labor Market

Let all firms be facing the same Cobb-Douglas production function

X = aL'S* (3)

where 0 < a < 1, 0 < B < 1, a + B = 1, and a > 0.

Such a linearly homogeneous production function rules out nonpure
competition rooted in economies of scale but not nonpure competition
rooted in product differentiation. So let each firm have found its
niche within a large group of rivals. The firm could sell more at a
lower price not matched by the rivals: their very multitude would make
the encroachment upon each of them so negligible that nobody would be
forced to follow suit. In that hypothetical case let all firms be
facing the same constant-elasticity demand function

P = bX*

where 0 > T] > -1 is the reciprocal of a Marshallian price elasticity of
demand. Our solutions will display the factor 0<l+t)<ltobe
thought of as a measure of the degree of competition in the goods
market. Under pure competition firms cannot control price, and the
equality sign holds: 1 + r\ =1. The less pure competition in the
goods market is, the lower the factor 1 + ti .

Nonpurely competitive firms optimize employment by equating the
money wage rate with the marginal-revenue productivity of labor:

w* (1 + r\)p££

Carry out the differentiation, rearrange, and write mark-up price

P = I ^k (4)

o(l + x\) X

Knowing the mark-up price (4) corresponding to a given money wage
rate, we know the real wage rate w/P and may express the demand for
labor as a function of it. Insert (3) into (4) and rearrange:

L = [aa(l + ii)]l/*(w/P)'1/*S <5>

As Lindbeck-Snower (1989: 373) observed, firms facing the same
production and demand functions as well as the same factor prices will
behave alike, i.e., decide on the same factor use (5) and (12), the same
physical output (3), and the same mark-up price (4). If so, (3), (4),
(5), and (12) are easily aggregated.

Facing aggregate demand for labor (5), how do unions respond?
Friedman's answer (1968) was his "natural" rate of unemployment to which
current labor-market literature adds institutional color: Lindbeck and
Snower (1986) and Blanchard and Summers (1988) distinguish between
"insiders," who are employed hence decision-making, and "outsiders," who
are unemployed hence disenfranchised. Let insiders accept the "natural"
employment rate X where 0 < X < 1. In other words, if L > XF insiders
will insist on a higher real wage rate. If

L=XF <6)

they will be happy with the existing one. If L < XF they will settle
for a lower one. We may think of the rate A. as a measure of the degree
of competition in the labor market: under pure competition unions
cannot control real wage rates, and the eguality sign holds, X = 1. The
less pure competition in the labor market is, the lower the rate X.

8
The real wage rate insiders will be happy with, given their
natural rate X of employment, might be called the "natural" one. Find
it by inserting (6) into (5) and rearranging:

w/P = ao(l + n) (\F)-*Sfi (7>

At a frozen capital stock S, then, labor can have a B percent
higher natural real wage rate by accepting a one percent lower natural
rate X of employment. Under rational expectations only random hence
unanticipated variations of the money supply can generate deviations of
actual from natural.

5. Capital Market

But in the long run physical capital stock S is not frozen but
optimized. Define, first, its physical marginal productivity

«-■§•»! (8)

and, second, its marginal-revenue productivity

(1 ♦ t\)kP

Let capital stock be immortal. Then at time t marginal-revenue
productivity (1 + t|)K(t)P(t) is marginal net return on capital stock,
hence taxed at the rate T. Aftertax marginal-revenue productivity of
capital stock is then (1 + r| ) ( 1 - T)K(t)P(t).

Let there be a market in which money may be borrowed at the
nominal rate of interest r. It will follow presently from our
definition (9) and our solution (30) that r is stationary. Let nominal
interest expense be tax-deductible at the rate T. Then money may be
borrowed at the aftertax nominal rate (1 - T)r. Let that rate be
applied when discounting future cash flows. As seen from the present
time t, then, aftertax marginal-revenue productivity of capital stock is
(1 + ti ) (1 - T)K(t)P(t)e"(1 " T)r(t " T). Define present gross worth of
another physical unit of capital stock as the present worth of all
future aftertax marginal-revenue productivities over its entire useful
life:

(t) ■ r (1 + n) (1 - DK(c)P(t)e-(1 " nr(t- T)

dt

Let the firm expect physical marginal productivity of capital
stock to be growing at the stationary rate gK:

10

K(t) = K(T)e*(t"T>

and price of output to be growing at the stationary rate gp:

Pit) = P(x)eaple't)

Insert these into the integral, define

p - (1 - T)r - (gK + gp) (9)

and write the integral as

(t) = J" (1 + r\) (1 - DK(T)P(T)e-«,(t-°

dt

Neither (1 + r\) , (1 - T), k(t), nor P(t) is a function of t,
hence may be taken outside the integral sign. Our gK, gp, and r were
all said to be stationary, hence the coefficient p of t is stationary,
too. Assume p > 0 and find the integral to be

k = (1 + ti) (1 - r)KP/p (10)

Find present net worth of another physical unit of capital stock
as its gross worth minus its price:

11

n ■ k - P - [(1 + i]) (1 - Dk/p - l]p

Optimized capital stock is the size of stock for which the present
net worth of another physical unit of capital stock equals zero, or

(1 + i|) (1 - T)K - p (X1>

Finally insert (8) into (11) and find optimized capital stock

S = P(l + r\) (1 - T)X/p (12)

Again, for optimized capital stock, firms facing the same
production and demand functions as well as the same factor prices will
behave alike, i.e., decide on the same factor use (5) and (12), the same
physical output (3), and the same mark-up price (4). If so, (3), (4),
(5), and (12) are easily aggregated.

In accordance with the definition (2) the growth of optimized
capital stock (12) is optimized investment

J - g^ = P(l + ti) (1 - Tig^/p (13)

What is p? As we shall see presently, our model will have the
solution (28) g = 0. Historically, for good measure, k has displayed

12
no secular trend. But if gK = 0 the definition (9) of p collapses into
the aftertax real rate of interest (1 - T)r - gp.

Keynes and Fisher are special cases of (11). If there is pure
competition, 1 + r\ =1, and taxation but no inflation, gp = 0, (11)
collapses into the Keynesian case k = r. If there is pure competition,
1 + r] =1, and inflation but no taxation, T = 0, (11) collapses into
the Fisherian (1896) case k = r - gp. But with nonpure competition,
inflation, and taxation all present, nothing less than (11) will do.

6. Consumption

Capital stock was assumed to be immortal, so we may ignore capital
consumption allowances and define national income as the market value of
physical output

Y - PX <14)

Let all such national income be taxed once and at the rate T.
Then tax revenue is

R = TY <15)

where 0 < T < 1,

13
Define disposable income as national income minus tax revenue:

y - Y - R (16)

Let consumption be the fraction c of disposable real income:

C = cy/P (17>

where 0 < c < 1.

7. Money

As a first approximation let government finance a deficit by
increasing the money supply. The government budget constraint then
collapses into

GP - R = gMM (18)

As a first approximation let the demand for money be a function of
money national income but not1 of the rate of interest:

14
D = mY (19)

where m > 0.

8. Equilibrium

As we saw, the labor market may not clear: there is a natural
rate of employment 0 < X < 1. But two equilibrating variables, i.e.,
the aftertax real rate of interest p and price P will clear the goods
and money markets, respectively:

X = C + I + G (20)

M=D (21)

We may now solve our system for its rates as well as its levels of
growth.

15
III. SOLUTIONS

1. Convergence of Growth Rates

Consider the growth rates g of joint factor productivity, gF of
available labor force, and gM of the money supply to be parameters, and
let all other growth rates be variables to be solved for. Consider our
natural rate of employment X a stationary parameter. Then insert (6)
into (3), take the growth rate (1), and find

9X = 9a + o-9P + $9S (22)

Insert (2), (14) through (19), and (21) into (20), rearrange, and
find the rate of growth of physical capital stock

g3 = [(1 - c) (1 - D - 9„™\XlS (23)

For the following reasons the square bracket of (23) is
stationary. First, the growth rate gM of the money supply was said to
be stationary. Second, all parameters c, m, and T were said to be
stationary. Consequently the growth rate (1) of the growth rate (23),
i.e., the rate of acceleration of physical capital stock is simply

16

9gsm 9x~ 9S

Insert (22) and write the rate of acceleration as

9gs = *(9j* + 9F ~ 9S) <24)

In (24) there are three possibilities: if g_ > ga/a + gf then

ggS < 0. If

9S = 9 J* + 9r <25>

then g - = 0. Finally, if gs < g /a + gF, then g _ > 0. Consequently,
if greater than (25) gs is falling; if equal to (25) gs is stationary;
and if less than (25) gs is rising. We conclude that gs must either
equal g /a + gF from the outset or, if it does not, converge to that
value.

Solow's (1956) convergence proof, then, survived our adding
government and nonpure competition. With the convergence proof secured,
we may easily find the corresponding values of other growth rates:
insert (25) into (22), recall that a + R = 1, and solve for the growth
rate of physical output

17
ffx = 9S <26>

Take the growth rate (1) of (7), insert (25) and (26), and solve
for the growth rate of the real wage rate

9V/P = 9 Jo- (27>

Take the growth rate (1) of (8), insert (26), and solve for the
growth rate of physical marginal productivity of capital stock

gK = 0 (28)

Insert (19) and (21) into (14), take the growth rate (1) of the
result, insert (25) and (26), and solve for the rate of inflation

9P = 9M ~ (9a/<* * gF) <29)

2 . Levels

We already have the solution for labor employed:

L=XF (6)

18
Insert (23) into (12) and solve for the aftertax real rate of
interest

p = P(l + r\) (1 - T)/H, where <30)

(l - c) (l - r) - g Mm

H ■ —

where g_ stands for (25). Next insert (3), (6), and (30) into (12) and
solve for physical capital stock

S = a^'H^'XF <31>

Insert (6) and (31) into (3) and solve for physical output

X = a1/aH*/a\F <32)

Insert (31) into (7) and solve for the real wage rate

w/P = a1/aa(l + ti)Hp/a <33)

Insert (19), (21), and (32) into (14) and solve for price

19

p = — T-4-, — (34>

a1/aH*/aXFm

Our levels (30) through (34) will be positive and finite if and
only if

H > 0, (35)

an inequality easily satisfied.

Our level (30) is stationary. Our levels (31), (32), (33), and
(34) are indeed growing at the rates (25), (26), (27), and (29),
respectively.

Divide (18) by Y, use (19) and (21), and solve for the deficit
share of money national income

GP - R (36)

= 9Mm

Rearrange (36), insert (15), and solve for the government-purchase
share of money national income

20
GP/Y = gMm + T (37)

Our levels (36) and (37) are stationary.

IV. SENSITIVITY OF SOLUTIONS TO MARKET STRUCTURES

1. Sensitivity of Levels to Nonpure Competition in Goods Market

Our factor prices (30) and (33) are in direct proportion to 1 + t|
hence are the lower the less pure competition is in the goods market.
No other level has any 1 + r\ in it.

Less pure competition, then, merely depresses factor prices but
has no effect upon factor use (6) and (31), physical output (32), or the
price of goods (34). As a result, less pure competition redistributes
income in favor of entrepreneurs at the expense of capitalists and
labor. Such redistribution is the result of a successful search for
niches — hence is a Schumpeterian incentive to such a search.

21
2. Sensitivity of Levels to Nonpure Competition in Labor Market

Factor use (6) and (31) and physical output (32) are in direct
proportion to X hence are the lower the less pure competition is in the
labor market. Price (34) is in inverse proportion to X hence is the
higher the less pure competition is in the labor market. No other level
has any X in it.

Such invar iance of factor prices (30) and (33) with 1 sheds new
light on union policy.

In the short run, i.e., at a frozen physical capital stock, labor
could have a higher natural real wage rate (7) by accepting a lower
natural rate X of employment. But in the long run, i.e., at an unfrozen
and optimized capital stock, labor can have no such higher real wage
rate: the natural rate X of employment is gone from our real wage rate
(33)1 The clue is scale. If both factor use (6) and (31) and physical
output (32) are in direct proportion to X, a lower natural rate X simply
reduces the entire economy to a lower scale. Here factor marginal
productivities remain the same. But the economy is accumulating
proportionately less capital stock and producing proportionately less
output, i.e., is impoverishing itself. Labor doesn't benefit. Nobody
benefits.

The time has come to turn to public policy.

22
V. SENSITIVITY OF SOLUTIONS TO PUBLIC POLICY

1. Sensitivity to Monetary Policy

Take partial derivatives with respect to the rate of growth of the
money supply of (6) and (29) through (37) and find:

P = 1 (38)

4^- = 0 (39)

°9M

-|£. = -?L£ > o (40)

d9„ 93 H

te = -1J11 < o (41)

dgM a gs H

4*- = -1-™ * < o (42)

dgH a. g3 H

23

jjg/g) = -£JLJ*/P < o (43)

d9„ * 93 "

4L = IjLl > o (44)

3[(gP-*)/Y] =m> Q (45)

d9M

d{f^ =m>0 (46)

d9n

2. Friedman's Dichotomy

Our growth-rate solutions (25) through (29) do display Friedman's
(1968) dichotomy between real and nominal variables: no growth-rate
solution for the four real variables S, X, w/P, or k has the rate of
growth of the money supply in it, and the growth-rate solution for the
nominal variable P does have the rate of growth of the money supply in
it.

But our level solutions (30) through (34) display no such
dichotomy: the level solutions for the four real variables p, S, X, and

24
w/P all have the rate of growth of the money supply in them, and so does
the level solution for the nominal variable P.

3. Sensitivity to Fiscal Policy

Take partial derivatives with respect to the tax rate of (6) and
(29) through (37) and find:

^=0 (47)

dT

%. - 0 <48>

dT

3£ = P(l * r\) 9Mm >

dT gs h2

jg a-AA-Z-£jg < o (50)

dT a g3 H

M = -lllii < o

(51)

25

d(w/P) = _ P 1 - c w/P Q (52)

if = P 1 - CP > 0
dT a g3 H

(53)

d[(GP - R)/Y]
dT

0 (54)

d(GP/Y) _ (55)

3r

4. Conclusions on Public Policy

Monetarist and New Classical supply-side equilibria were as static
as the Keynesian demand-side equilibrium had been: capital stock
remained frozen rather than being optimized. At such frozen capital
stock and a "natural" rate of employment, physical output would be
immune to monetary and fiscal policy.

Did our dynamization of supply-side equilibria make any
difference?

It did indeed. Consider a higher gM or a higher T — both
permitting a larger public sector according to (46) and (55). Our

26
long-run unfrozen and optimized capital stock and output were the
functions (31) and (32), respectively, of our policy instruments gM and
T. How would (31) and (32) respond to a higher gM or a higher T?

Certainly not in a Keynesian manner: they would not be up.
Certainly not in a monetarist manner: they would not stay the same.
According to our (41) and (42) or (50) and (51) they would be down.
According to (43) and (52) the real wage rate would also be down. What
brought such calamities about was a higher aftertax real rate of
interest according to (40) and (49).

A truly expansionary policy, then, should lower, not raise gM and
T. But whereas a lower gM would also lower inflation and the deficit
share according to (38) and (45), a lower T would leave inflation and
deficit share the same according to (47) and (54).

5. What Survived?

Two essential propositions of a supply-side equilibrium survived
our nonpure competition and our dynamization. The first is the immunity
of employment (6) to public policy: according to our (39) and (48) such
policy cannot affect employment.

The second surviving proposition is the existence of a price
mechanism. We have indeed treated price as an equilibrating variable

27

and have solved in (29, for its growth rate and in ,34) for its ievei.
Both solutions were found to be sensitive to public policy.

HB.1-6

28

FOOTNOTES

1A zero elasticity of the demand for money with respect to the
nominal rate of interest was accepted by an early Friedman (1959) but
not by a later one (1966).

2The economic meaning of the inequality (35) is simply that
crowding-out must never be complete. Reality easily satisfies such an
inequality. Let 1 - c = 0.08 and m = 1/5. Then at a United-States-like
gM = 0.06 the tax rate T could rise to 0.85 before violating (35). Or
at a United-States-like T = 1/4 the rate of growth gM of the money
supply could rise to 0.3 before violating (35).

29

REFERENCES

Blanchard, Oliver J., and Summers, Lawrence H. , "Hysteresis and the

European Unemployment Problems," in Cross, Rod (ed.) Unemployment .
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Fisher, Irving, "Appreciation and Interest," Publications of the
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Friedman, Milton, "The Demand for Money: Some Theoretical and Empirical
Results," J. Polit. Econ., Aug. 1959, 67, 327-351.

, "Interest Rates and the Demand for Money," J. Law Econ. ,

Oct. 1966, 9, 71-85.

, "The Role of Monetary Policy," Amer. Econ. Rev., Mar.

1968, 58, 1-17.

Lindbeck, Assar, and Snower, Dennis J., "Wage Setting, Unemployment, and
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235-239.

30
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Rev. . May 1989, 79, 370-376.

Lucas, Robert E., Jr., "Expectations and the Neutrality of Money," J.
Econ. Theory, Apr. 1972, 4, 103-124.

Sargent, Thomas J., "Rational Expectations, the Real Rate of Interest,
and the Natural Rate of Unemployment," Brookings Papers on
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Sargent, Thomas J., and Wallace, Neil, "'Rational' Expectations, the

Optimal Monetary Instrument, and the Optimal Money Supply Rule,"
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Solow, Robert M. , "A Contribution to the Theory of Economic Growth,"
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